ICF13 Beijing (China) 2013 Vol. A
1 “Sir Alan gave but never broke Like steel he studied and like oak, He brought up science by his hand The royal smith kept on the brand!” The invariant integral: some news Genady P. Cherepanov Honorary Life Member, the New York Academy of Sciences, USA (Elected on Dec. 8, 1976 together with Linus C. Pauling and George Polya) genacherepanov@hotmail.com Abstract. Earlier, this author introduced the invariant integral as a general mathematical tool for solving the physical problems based on conservation laws, without using partial differential equations, similarly to the calculus of variations. In this paper, the invariant integral was introduced for cosmic, gravitational, electromagnetic, and elastic fields combined. In a particular case of the united cosmicgravitational field, from the corresponding invariant integral the force F acting upon point mass m from point mass M and from the cosmic field was derived : . Here: G is the gravitational constant, Λ is the cosmological constant, and R is the distance between the masses. The first term provides Newton’s gravitation and the second term the cosmic repulsion. This force was used to build an elementary non-relativistic cosmological model of Universe and estimate the size of Universe as well as explain the accelerated expansion of Universe recently observed by astrophysicists. The orbital speed of stars in galaxies was found out to be constant and equal to about 250 km/s. Keywords: invariant integral, cosmic and gravitational field, interaction force, expansion of Universe 1. Introduction The integrals which are invariant with respect to the integration contour or surface provide a way to write down the laws of conservation of energy, mass, momentum and so on. From them, one can derive the differential equations as the local representation of the same conservation laws. However, the invariant integral approach is more powerful because it allows one to also deal with the field singularities where the differential equations have no meaning. In 1967, using the energy conservation law, this author derived the main invariant integral for elastic and inelastic materials and introduced it into fracture science [1]. In this approach, the invariance of the integral with respect to any integration paths followed from the energy conservation law, so that this fact seemed to be trivial and was not discussed. Particularly, the characterizing index of power-law hardening materials and the similarity were found out, which constituted the basis of the later HRR approach. In 1968, while studying strain concentration by notches and cracks, Jim Rice
2 utilized this integral to prove its invariance using the divergence theorem. (For this famous work, Jim was awarded several medals and prizes). In 1972, John Landes and Jim Begley, not familiar with paper [1], re-introduced the invariant integral into fracture mechanics. At about that time this author discovered that in the case of elastic materials his integral coincided with that of Jock Eshelby introduced earlier into the theory of point defects in crystal lattices. Since 1968, Jock started on actively working in fracture mechanics, too. I can’t help but recall some events of that time related to paper [1]. Since 1959 my basic scientific interests have been connected with crack growth. However, up to 1964 when I became the youngest Doctor of Science in USSR, my most significant publications were done on the problems of mechanics with unknown boundaries, including the elastic-plastic, local buckling, and contact problems. It’s on these problems I earned all my degrees, although fracture has always been my main subject. At that time in USSR, this area was monopolized by unscrupulous, powerful figures close to KGB who vetoed approaches different from their own. Still and all, in 1965 I decided to leave underground and submitted a Russian manuscript entitled “On crack propagation in continuous media” into the journal “Prikladnaia Matematika I Mekhanika” ( PMM or Journal of Applied Mathematics and Mechanics, JAMM). However, the publication of the paper was blocked up by the authorities and it was kept in the portfolio of the journal for two years although my earlier, less significant papers used to come out within half a year. And yet, my luck was in because the Editor-in-Chief Leo Galin, even without knowledge of the paper subject, took over the responsibility and published my paper, much mutilated though by censors for two years. At last, it came out by May 1, 1967. Later, I and brave Professor Galin paid a heavy price for this sin. By the way, after the Soviets launched the first sputnik in 1957, the urgent airmail delivered fresh issues of PMM to the best US university libraries within several days. Since that time, I tried to show that fracture mechanics is a legitimate branch of theoretical physics and my invariant integral can be used as an efficient mathematical tool for solving singular physical problems far beyond fracture mechanics (see my book [2] written in1969 but published in Russian only in 1974 and in English later, in 1978, by McGraw Hill ). However, these ideas were poorly understood. Hopefully, what follows below can make a difference. 2. General case Let us consider stationary processes in elastic dielectrics, with taking account of cosmicgravitational and electromagnetic forces. In this case, the main invariant integral can be written as follows [3 - 5] + , , , =1,2,3 (1)
3 Here: any closed surface in the space of Cartesian coordinates; the gravitational and cosmological constants respectively; the potential of united cosmicgravitational field; the vector components of electromagnetic field; the potentials of electromagnetic and elastic fields respectively; elastic displacements and stresses; the orts of the outer normal to Vector represents the driving force of field singularities inside Σ, which equals the work spent to move the singularities for unit length. If all then there are no singularities inside Σ; in this case the basic differential equations inside Σ can be derived from the invariant integral (1), e.g. the Maxwell equations, the equations of elasticity theory and gravitation. Moreover, Eq. (1) allows one to derive the interaction laws for any particular cases, e.g. Newton’s law of gravitation, Coulomb’s law for electric charges, Ampere’s law for electric currents, Joukowski’s equation for wing lift, Irwin’s law for crack driving force, PeachKoehler’s law for dislocation driving force, Eshelby’s law for point inclusion driving force, as well as many new ones [2-5]. For example, the interaction force of two electric charges moving in a dielectric medium along the common symmetry axis at speed was found to be equal to [3-5] (2) Here: the distance between the charges in the proper reference frame; the dielectric constant; the speed of light in vacuum and medium respectively. For , Eq. (2) represents Coulomb’s law. If the force applies only to the rear charge and the force’s sign changes. Eq. (2) plays the main part in electron mode fracture [3, 5] by powerful electron beams. Taking account only of two last terms in the right-hand part of Eq. (1) provides the original invariant integral which is the basis of modern fracture mechanics [1-5]. 3. Cosmic-gravitational field In what follows we consider the united cosmic-gravitational field defined by the invariant integral as follows (3) In Eq. (3), the first term describes the flux of gravitational energy through the closed surface the second term the work of field tractions on and the third term the flux of cosmic energy through In the present non-relativistic approach, the cosmic energy can, probably, be
4 interpreted as “Dark Energy” , but we refrain from using this notion here. By ignoring the third term in Eq. (3) we arrive at the classical model of gravitational field [3-5]. Using physical dualisms the cosmological and gravitational constants can be written in other units. Let us put in Eq. (3). It means there are no field singularities inside In this case, by using the divergence theorem one can easily transform the surface integral in Eq. (3) into the volume integral and derive the following equation for which is valid at any point inside (4) From Eq. (4), it follows that represents the density of anti-gravitational matter (negative mass) which is uniformly distributed everywhere As a matter of fact, this is the physical interpretation of the cosmic field in the present model. For example, in this model the space volume which is equal to the volume of our Earth, contains of the antigravitational matter so that its density times less than the mean density of Earth. Now, suppose there is a point mass at a certain point O inside Using the invariance of the integral in Eq. (3) with respect to , one can shrink and turn it into a small sphere over point O . Then, applying the -integration procedure provides the following equation for the force upon mass [3] (5) Here, is the th component of the field intensity vector at point O, when there is no mass at point O. The physical nature of the cosmic force as well as gravity remains unclear despite the success of general relativity and numerous other theories. 4. Interaction force Let us find the interaction law of two point masses in the cosmic-gravitational field. Let mass be concentrated at point . Solving Eq. (4) provides the following field created by this mass (6) The intensity of this field at point is equal to (7) From Eq. (5) and Eq. (7), it follows that force upon mass at point directed along the axis is equal to . (8) Also, by analogy force upon mass at point is equal to
5 (9) Eqs (8) and (9) provide the interaction law of two point masses in this model. The first term describes Newton’s gravitation/attraction, and the second term the cosmic repulsion. The latter does not depend on the opposite mass that plays the only role of a trigger and gauge; and so, the latter does not follow the Newton’s law that action equals counteraction. When the distance increases, the gravitation tends to zero while the repulsion tends to infinity. Let us study some problems for two point masses. Two free masses in the cosmic-gravitational field. Let free masses on the x-axis be acted upon by forces of Eq. (8) and Eq. (9) where ( are the coordinates of corresponding masses movable along the - axis). The distance between the masses satisfies the equation . (10) If at the initial moment of time when then the masses move one towards the other until they collide. If at the initial moment of time when then the masses move apart one from the other until they disconnect. The solution to Eq. (10) is as follows ( where C is defined by initial conditions). (11) One mass is fixed and the other is free. Suppose mass is fixed at the coordinate origin and mass is free to move along the x-axis. In this case let us take into account the relativistic dependence of the latter mass on its velocity. (Yet, the introduced cosmic-gravitational field does not satisfy the special relativity). In this case the velocity of the free mass can be written as follows ( where C is defined by initial conditions) (12) Here c is the speed of light. According to the present model, the cosmic field describes the intrinsic geometrical property of material space for accelerated self-expansion. 5. The cosmological system Let us apply the introduced cosmic-gravitational field to cosmology. First, consider a finite system of any number of point masses in a finite volume of the 3D Euclidian space. Designate by the maximum distance between any two masses, and by the total mass of the system. The cosmic field attached to this system is inside the sphere of diameter According to Eqs (8) and (9) the dimensionless number
6 characterizes the ratio of repulsion force to that of attraction and, hence, this number characterizes also the global behavior of this system which depends on its scale. Evidently, when we can ignore the cosmic energy and its repulsion effect, and when we can ignore the gravitational energy and its attraction effect. From Eq. (10), it follows that a system of two masses expands and disappears, when and the system exists, when . Let us estimate the value of for some astronomical objects. Our solar system: Sun, Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto. The latter is distant from Sun, so that we can take (about the mass of Sun). From here, it follows that for our solar system. And so, there is no way to observe the cosmic field from any planetary observations, although the cosmic field makes the eccentricity of planetary orbits a little bit bigger by an undetectably small amount. Milky Way. Our galaxy Milky Way consists of more than 200 billion stars which rotate around the galaxy center where there is a Black Hole. Milky Way has a shape of a flat pancake having thickness , radius and mass about . From here, it follows that for Milky Way. And so, even in the scale of the galaxy it is, probably, impossible to measure the effect of the cosmic field. Our Universe. Our Universe consists of more than 100 billion galaxies packed in clusters and super-clusters, each of many millions of galaxies. Our Universe’s fractal dimension is about 2.2 so that it resembles a flat pancake, too. According to some recent estimates for our Universe, we have , and . For typical super-clusters, we have and . And so, it is evident that the effect of the cosmic field can be observed and estimated only from astronomical observations of very distant objects that are close to the edge of our Universe, i.e. to the time billion years since the Bing Bang has happened. Let us compare the gravitational force of attraction of Earth to Sun and the cosmic force of repulsion of Earth from Sun. The first one is equal to while the second Now, we present an elementary, non-relativistic model of our Universe and estimate its size as follows. Suppose our Universe of mass is homogenous so that a gravitational probe mass is acted upon by the gravitational force of attraction to the center of Universe and by the cosmic force of repulsion from the center of Universe where is the distance of the mass from the center of Universe. We define our Universe as a closed community of gravitational masses in the unbounded cosmic field. From here, it follows that the probe mass goes away and leaves Universe if where
7 ( ) (13) Hence, we can come up with the conclusion that represents the radius of Universe in this model which for provides i.e. comparatively close to the prediction of the Lambda-CDM model. If the probe mass is inside the homogenous Universe (i.e. , then the total force upon the mass resulting from the gravitational and anti-gravitational matter is zero so that, in average, any domain inside Universe experiences no contraction and no expansion. Moreover, the total mass of Universe is equal to zero. The gravitational matter is concentrated in many moving clots inside Universe while the anti-gravitational matter of the cosmic field is uniformly distributed everywhere. From here, it may be assumed that Universe is a gigantic fluctuation created from nothing (i.e. something which energy-mass was zero). However, the Universe is open so that, eventually, some gravitational masses on the edge move out and go away, that is they leave Universe forever. Their place in Universe is, then, taken by some insiders. This is the way how Universe expands and, evidently , this expansion is accelerated due to the arising and growing imbalance of gravitational and anti-gravitational forces inside Universe. The incorporation of the introduced cosmic field into the framework of general relativity seems to be impossible but it is quite plausible for the field theories using non-metric theories of gravity. 6. Orbital speed of stars in spiral galaxies The orbital speed of a planet in our solar system determined by equilibrium of inertia force to that of gravitation equals where is the mass of Sun and is the distance of the planet from Sun. And so, this speed decreases tending to zero when the distance grows. Because in Milky Way and other galaxies, the orbital speed of stars rotating around the center of Milky Way has, seemingly, to be described by a similar law. However, it is not. The orbital speed of stars in galaxies appears to be independent of the distance to the center and equal to about ( for our Sun, ). This paradox produced some theories. For example, the well-known MOND theory accepts that the inertia force upon a star in a galaxy is directly proportional not to the acceleration (equal to in the case of uniform circular motion) as follows from the Newton-Galileo mechanics, but to the square of acceleration. Certainly, this approach makes the orbital speed independent of Another school of thought considers Dark Energy responsible for this paradox.
8 Meanwhile, the structure of Milky Way and other spiral galaxies provides a simple explanation of the paradox using the classical Newton-Galileo mechanics and Newton’s law of gravitation which is the right approach because in the scale of galaxies so that the cosmic field can be ignored. The gravitational matter of Milky Way is uniformly distributed along the logarithmic spirals having the common pole at the center of Milky Way. These spirals are welldocumented. The length of an arc of a logarithmic spiral equals where is the constant angle between the radius-vector of a point on the spiral and the tangent to the spiral at the point while are the distances between the pole and the ends of the arc. The arc length is directly proportional to the distance of a point from the center of the galaxy when Mass inside of the galactic disc of radius is also directly proportional to because such proportionality remains for any number of spirals. In other words, where is a certain structural constant of the galaxy. The force of attraction of a star of mass to the center of the galaxy is equal to . This force is balanced by the inertia force of Galileo-Newton. From here, the orbital speed of stars is as follows (14) For Milky Way and . This value is close to astrophysical data. About the same value of the orbital speed of stars has been observed in all galaxies. It means that the density of gravitational matter in all galaxies obeys the following general law (15) Here, is the mass of gravitational matter inside unit area of the galactic disc, is the distance from the center of the galaxy, and is the universal galactic constant. And so, the matter density is infinite at the center which is the galaxy’s Black Hole. The logarithmic spiral is the only spiral that corresponds to this general law, see Eq. (15). 7. The Einstein equivalence principle and no annihilation paradox The general relativity is based on the equivalence principle which says that the gravity is the inertial force in the curved space-time. The cosmic-gravitational field does not obey this principle. We illustrate the difference using the following simple example. Let a mass , being fixed to a point by an inextensible string of length , rotate at a constant speed around the point. In absence of cosmic field, the inertia force is balanced by the extension force in the string. Replacing the action of this string by a gravitational mass placed at the center of rotation, such that , we come up with the equivalence principle. (It can be also formulated as the equity of inertial mass to gravitational one).
9 Now, imagine a slightly “pliant” string, which cross-section diameter being directly proportional to the square root of the distance from the rotation center and which strain inversely proportional to stress in the string, with a very small proportionality coefficient. In this case, substituting the increased value of in the above balance equation provides (16) Here, is the undeformed string length which is different from the real one by a small value that cannot be detected in the scale of solar system. In presence of cosmic force, the term imitates the cosmic force and Eq. (16) a modified equivalence principle. The no annihilation paradox of the present model of cosmic-gravitational field can be understood only after we get known the physical nature of cosmic field. Such a remark is valid also for many non-metric theories of gravity. References [1] G. P. Cherepanov, Crack propagation in continuous media (in Russian), Prikl. Mat. I Mekh. (PMM), v. 31, n. 3, (1967), 476-488. The English translation in: FRACTURE: A Topical Encyclopedia of Current Knowledge (G. P. Cherepanov, Ed.), Krieger Publ., Melbourne, 1998, 41-53. [2] G. P. Cherepanov, Mechanics of Brittle Fracture (in Russian), Nauka, Moscow, 1974. The English translation: McGraw Hill, New York, 1978. [3] G. P. Cherepanov, Fracture Mechanics (in Russian), IKI publ., Izhevsk-Moscow, 2012. [4] G. P. Cherepanov, Some new applications of the invariant integrals of mechanics (in Russian), Prikl. Mat. I Mekh. (PMM), v. 76, n. 5, (2012), 823-849. The English translation: in JAMM. [5] G.P. Cherepanov, Methods of Fracture Mechanics: Solid Matter Physics, Kluwer, Dordrecht, 1997.
13th International Conference on Fracture June 16–21, 2013, Beijing, China -1- The Characterization of Dominating Region of Fracture (process region) around a Crack tip Based on the Concept of Mechanical Similarity and Atomic Mechanics A. Toshimitsu Yokobori, Jr. 1,*, Yoshiko Nagumo 1, Takahiro Yajima 1 Toshihito Ohmi 1 1 Department of Nanomechanics, Tohoku University, Sendai, 980-8579, Japan * Corresponding author: yokobori@md.mech.tohoku.ac.jp Abstract Brittle fracture at a crack tip is considered to be caused within a specified local region, that is, process region. However, the determination of this scale has not yet been theoretically clarified. The difficulty of this determination will be due to the wide range scale analysis from the range of nano scale (atomic scale) to macro scale (crack size scale). In this paper, on the basis of the atomic mechanics using super atom and hybrid method of the fractal concept which concerns self-similarity with the proposed analysis of scale projection from macro to nano scales, the disturbed region of atom arrangements around a crack tip were clarified under the local stress field by crack and dislocation. This disturbed region was related to fracture dominating region (process region) which is in good agreement with the scale obtained from experimental consideration by R. Ritchie. Keywords Process region, Mechanical similarity, Atomic mechanics, Brittle fracture 1. Introduction The fracture criterion of materials is considered to be a condition that fracture occurs when a specified mechanical condition is satisfied in the local damage region around a crack. As the criterion, the energy and the local stress conditions [1-3] in this region or the hybrid theory which is taken into account for both conditions [4] have been proposed. There are corresponding local damage regions in brittle fracture, fatigue fracture and creep fracture, respectively. Each scale of these local damage regions is different respectively. For example, each local damage region is considered to correspond with the plastic region [5] for fatigue fracture and creep damage region [6] for creep fracture. For brittle fracture, the end region or process region are considered to be the dominating region of fracture [7]. However, its physical meaning has not yet been clarified. Under this condition, determination of this local region using the experimental fracture toughness has been conducted and the scale of this region is considered to be that of several grains [8]. The process region is considered to be a disordered region of atomic arrangement [3] which concerns fracture of atomic bond. This analysis requires the effects of crack, dislocations and atoms of which scales are order of mm, μm and nm on local stress field around a crack or dislocation groups. Since it is necessary to construct two dimensional atomic arrangements to realize a modeled crack or dislocations, conduction of this analysis by atomic mechanics alone requires large scale and high accuracy numerical analyses. Under this background, authors aim to show the possibility of conducting the analysis of determination of local dominating region of brittle fracture by convenient scale of numerical analysis such as PC level analysis to conduct various calculations conveniently. The concept of this analysis is as follows. (1) The macro and micro local stress fields such as a crack and a dislocation are introduced as mechanical boundary conditions of the region of analysis. It causes the elimination of the necessity of the establishment of two dimensional atomic models, which gives the validity of conducting one dimensional analysis. (2) By utilizing property of potential system of atomic force, instead of using actual atoms array, super atoms array is considered since the mechanical similarity is valid between them. Super atom
13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- has mechanical equivalence with several actual atoms. Under the boundary condition of (1), an array of these super atoms was considered and a convenient mechanical analysis which obtains mechanical equilibrium positions of super atoms was conducted. (3) Furthermore, these super atoms were sequentially dispersed under the mechanically equivalent state. Consequentially, the number of these atoms was increased and at each stage of the corresponding scale of these atoms, the mechanical equilibrium positions of these atoms were numerically analyzed. (4) The characteristic of self-similarity of atom arrays was investigated based on the fractal analysis through each stage of the scale of super atoms. Super atoms were sequentially dispersed and the mechanical equilibrium positions of these atoms were numerically analyzed. (5) The numerical analyses were conducted up to the scale stage of the super atomic array which shows the characteristic of self-similarity. From this scale stage of super atoms, disordered region of atomic arrays around the crack tip or dislocation with the actual scale of atomic array was predictively calculated and this region was defined as the process region of fracture. Using this theory, the characterization of disordered region in the macroscopic local stress field around the crack tip becomes possible conveniently and with high accuracy. It enables us to theoretical prediction of the process region which is the dominant region of brittle fracture without the aid of experimental results. 2. The Model and Method of Analysis 2.1. The Model of Analysis The concept of super atoms and dislocations which represent mechanically equivalent several actual atoms and dislocations, that is, the concept of image atoms and dislocations were used. Since stress field around a dislocation and interactive force between atoms are conservative, mechanical similarity on scale will be valid. Therefore, similar equations of stress field and interactive forces of dislocation and atoms of which values of intensity are different are adopted for super atoms and dislocations. At each scale stage of super atoms, using equations of interactive forces which have corresponding intensity, the equilibrium positions of super atoms were numerically analyzed. The flow of analysis and the concept of mechanical equivalence between super and actual atoms were shown in Fig. 1. With increment increase in the number of atoms, interactive forces between these atoms were dispersed and the equilibrium positions of atoms in the array were numerically analyzed respectively. The fractal analyses were conducted for the morphology of distribution of atomic density in the array at each scale stage of super atoms and the existence of self-similarity of the atoms array was investigated through each scale stage. When the characteristic of the self-similarity appears through each scale, the projection method is applied to the sequential changing characteristics of the morphology of distribution of atomic density to predict the equilibrium distribution of actual atomic arrays by considering similarity of the changing characteristic being kept due to the existence of the self-similarity. The dominating local region of brittle fracture was predicted by this method and results obtained was compared with previous results [8].
13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- Figure 1. Dispersion method from super atoms to actual atoms under the mechanical equivalence (Scaling from macro to micro under the conservative mechanical system) 2.2. Basic Equation In this analysis, the number of N of super atoms were placed between a crack and a super dislocation which represents dislocation groups. The equations of interaction forces exerted on super atoms were mechanically considered to be symmetry with those exerted on actual scale atoms [9]. The mechanical model used for this analysis was shown in Fig. 2. A crack tip exists at the site of x = 0 and shearing stress was exerted parallel to the crack that is mode II condition. The super dislocation exists at the site of x = d. The number of n of super atoms were placed the same scale interval. In this analysis, to avoid the jump out of atoms from this region during the process of analysis of obtaining the equilibrium positions of atoms between the crack and the super dislocation, fixed atoms were placed at the site of the crack tip and the super dislocation respectively. Local stress fields by the crack and super dislocation were given by boundary conditions in this analysis. The interaction forces between these local stress fields and fixed atoms were neglected. The interaction force exerted on each super atoms due to the crack, super dislocation and other super atoms in the array was given by Eq. (1), ( ) ∑ + ≠ = ∗ − − + ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = 1 0 7 13 * 2 6 12 4 N j i j i d i II j i j i i x x nA x K x x x x I π σ σ σ ε (1) Where i=1~(N-1), ε : constant with dimension of energy, σ : constant with dimension of length, KII：stress intensity factor of mode II which concerns local stress of the crack tip, xi：the position of the ith atom, n : intensity of the super dislocation, A*：intensity of an isolated actual dislocation. The first and second terms of right hand side of Eq. (1) were interactive forces due to other super atoms exerted on the ith atoms. The third and the forth terms of right hand side of Eq. (1) were interactive forces due to local stress fields of the crack and super dislocation exerted on the ith super atom. Figure 2. The mechanical equilibrium model of a crack, a super dislocation and super atoms Crack Super Atoms Super Dislocation 0 1 2 3 4 N-1 N N+1 ･･････ x=0 x=1 Self-similarity
13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- 2.3. The algorithm of mechanical dispersion of atom groups The value of x+ i,n is taken as the non-dimensional variable given by Eq. (2). σ is defined as l. Concerning the dispersion up to the scale of atom, 1/(βN)13 and 1/(βN)7 was exerted to the first and the second terms of Eq. (1) respectively to decrease and disperse interactive forces between super atoms with increase in the number of atoms. By converting Eq. (1) using Eq. (2), the non-dimensional representation of Eq. (1) was given by Eq. (3). That is, 1/(βN) is an adjusting parameter of the reasonable intensity of potential field corresponding with each scale of atoms. l x x i n i n , , = + , l =σ (2) Where, +x is non-dimensional position of atom, β is a dispersion coefficient of super atoms. l is a representative length and it equals to value of σ. ( ) ( ) ( ) ( ) ( ) ∑ ≠ = + − + ∗ + − + − + − + − + − ∗ − − + ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ − − − = N j i j i n d i n II j n i n j n i n i n x x nA x K l x N x x N x I 1 , 1 , 1 7 , 1 , 1 7 13 , 1 , 1 13 , 2 1 6 1 12 4 π β β ε (3) 2.4. Method of analysis On the basis of the basic equation, to obtain equilibrium positions of super atoms, following two methods were adopted. 2.4.1. Direct method Equation (2) represents forces exerted on each super atom. When value of Ii * is not zero, each super atom moves toward the corresponding equilibrium position dominated by Eq. (4). ∗ = i n i n I dt d x M , 2 , 2 (4) Where n is the step number when corresponding atom moves step by step toward the equilibrium position under the Eq. (3). By Eq. (4), velocity of each atom at the time incremental value, Δt is given by Εq. (5). Therefore, moving distance during the time incremental value, Δt is given by Εq. (6). Using Εq. (6), the position of each atom at the time of (t+Δt) is given by Εq. (7). Where M is an imaginary mass and for convenience from the view point of calculation, M was taken as unity. , 1 * , , − = Δ + i n i n i n V I t V (5) x V t i n i n Δ = Δ , , (6) i n i n i n x x x , , , 1 = +Δ + (7) This method [9] can be applied to any cases of moving distance of atoms, however the calculation time of numerical analysis becomes longer as compared with Verlet method. 2.4.2. Verlet method On the basis of Taylor expansion, Verlet method was proposed to obtain numerical solutions for the type of Eq. (4) with high accuracy and short calculation time. This method is mainly applied to molecular dynamics. This method is summarized as follows. On the basis of Taylor expansion, finite difference representation of two order derivative was given by Eq. (8).
13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- ( ) ( ) ( ) ( ) 2 2 2 2 dt d x t t x t t x t x t +Δ + −Δ = + Δ (8) Using Eq. (8), xi,n+1 is given by Eq. (9). ( ) ∗ − + = − + Δ i n i n i n i n t I x x x , 2 , 1 , , 1 2 (9) In this paper, these two methods were adopted to conduct this numerical analysis and results obtained by these methods were compared from the view point of accuracy, the number of atoms which can be calculated and calculation time of numerical analysis. The converged conditions on the equilibrium positions of atoms were given by Eqs. (10) and (11). 9 , , , 1 10− + < − i n i n i n x x x (10) 4 1 * , 10− = < ∑ N I N i i n (11) The center position of atomic distance and atomic density and the changing rate of atomic density to the initial atomic density were given by Eqs. (12)-(14). 2 1 i i i x x X + = − (12) 1 1 − − = i i i x x d (13) ,0 ,0 i i i i d d d D − = (14) The center position of atomic density was shown in Fig. 3. The positive value of D means that the distance between neighboring atoms expands due to local stress field. The region with negative value of D was defined as the disordered region of atomic array and it is related to the process region which dominates brittle fracture [7, 8]. Conditions of analyses were shown in Table 1. Figure 3. The definition of the center position of atomic distance Table 1. Conditions of analyses Condition1 Condition2 Condition3 KII (MPam1/2) 20 40 20 nA*(MPa) 0.049 0.049 0.1 N 10~1000 (Direct method) 10~3000 (Verlet method) xi xi-1 The center position of atomic distance
13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- Analyses were conducted for the following cases. (1) The case of local stress by the crack and the super dislocation being equivalent at the center position of (0, 1). (Condition 1) (2) The case of local stress by the crack being two times larger than that of super dislocation at the center position of (0, 1). (Condition 2) (3) The case of local stress by the crack being half of that of super dislocation at the center position of (0, 1). (Condition 3) Since the stress singularity of the crack against the distance from the crack tip (r-1/2) is smaller than that of super dislocation (r-1), the high stress region of the former is wider than that of the latter. 2.5. Fractal analysis based on Box counting method In this paper, fractal analysis was adopted to estimate the self-similarity of the distributed characteristics of D based on Box counting method. 3. Results of Analysis 3.1. The comparison of two methods and the characteristics of changing rate of atomic density, D The effects of atomic scale and number on the characteristics of D defined by Eq. (14) were shown in Figs. 4(a), (b) and (c) under three conditions. Results obtained by both of the direct and Verlet methods showed the same characteristics of atomic distributions that atomic density increase at the center position of analysis region, (0, 1) and decrease at positions of both ends, that is the crack tip and the super dislocation. The direct method can be applicable for any case of the moving distance of atom, however the latter is based on the Taylor expansion and the small moving distance is required. Furthermore, the calculating time of the former is much longer than that of the latter as shown in Fig. 5. In the region of D having a negative value as shown in Figs. 4 (a), (b) and (c), the breaking stress of atomic bond was applied. Therefore, this region is considered to be a dominating region of fracture. As shown in Figs. 4(a), (b) and (c), with increase in the effect of the crack or the super dislocation, correspondingly, the occurrence of the region of D having negative value near the crack or dislocation becomes typical. Normalized distance, X The changing rate of atom density, Di N=250(Direct) N=250(Verlet) N=500(Direct) N=500(Verlet) N=1000(Direct) N=1000(Verlet) 0 0.2 0.4 0.6 0.8 1 -0.08 -0.06 -0.04 -0.02 0 Normalized distance, X The changing rate of atom density, Di N=250(Direct) N=250(Verlet) N=500(Direct) N=500(Verlet) N=1000(Direct) N=1000(Verlet) 0 0.2 0.4 0.6 0.8 1 -0.08 -0.06 -0.04 -0.02 0 Normalized distance, X The changing rate of atom density, Di N=250(Direct) N=250(Verlet) N=500(Direct) N=500(Verlet) N=1000(Direct) N=1000(Verlet) 0 0.2 0.4 0.6 0.8 1 -0.08 -0.06 -0.04 -0.02 0 (a) Condition 1 (b) Condition 2 (c) Condition 3 Figure 4. Plots of changing rate of atomic density D vs. normalized distance on each condition
13th International Conference on Fracture June 16–21, 2013, Beijing, China -7- Number of super atoms, N CPU TIME.[sec.] Direct Method Velret Method 10 100 1000 1 10 100 1000 10000 100000 Figure 5. Plots of calculating time vs. number of super atoms on each method 3.2. Fractal analysis of the distribution of D values On the basis of Box counting method, fractal dimensional values, FD were calculated and they were plotted against the number of super atoms as shown in Figs. 6 (a), (b) and (c). These results show that FD takes a specified saturated constant value when the number of super atoms is larger than Nc = 250. Number of super atoms, N Fractal dimensionality, FD Direct Verlet Nc=250 1 10 100 1000 10000 1 1.05 1.1 1.15 1.2 Number of super atoms, N Fractal dimensionality, FD Direct Verlet Nc=150 1 10 100 1000 10000 1 1.05 1.1 1.15 1.2 Number of super atoms, N Fractal dimensionality, FD Direct Verlet Nc=100 1 10 100 1000 10000 1 1.05 1.1 1.15 1.2 (a) Condition 1 (b) Condition 2 (c) Condition 3 Figure 6. Plots of fractal dimensional value vs. the number of super atoms on each condition 3.3. Determination of dominating region of fracture using projection method To predict the dominating region of fracture, the region where a value of D takes negative value is defined as the disordered regions of super atoms. As shown in Fig. 7, they were characterized by L at the side of the crack and G at that of super dislocation. The changing characteristics of L and G were plotted against the number of super atoms as shown in Figs. 8 (a), (b) and (c). Both of characteristics of L and G were found to take constant values, respectively in the region where the distribution of D shows fractal characteristics.
13th International Conference on Fracture June 16–21, 2013, Beijing, China -8- Normalized distance, X The rate of atom density change, Di L G 0 0.2 0.4 0.6 0.8 1 -0.08 -0.06 -0.04 -0.02 0 Figure 7. Definition of disordered regions of super atoms Number of super atoms, N Length of disordered region of super atoms, L,G L(Direct) L(Verlet) G(Direct) G(Verlet) 10 100 1000 10000 0 0.1 0.2 0.3 0.4 0.5 Number of super atoms, N Length of disordered region of super atoms, L,G L(Direct) L(Verlet) G(Direct) G(Verlet) 10 100 1000 10000 0 0.1 0.2 0.3 0.4 0.5 Number of super atoms, N Length of disordered region of super atoms, L,G L(Direct) L(Verlet) G(Direct) G(Verlet) 10 100 1000 10000 0 0.1 0.2 0.3 0.4 0.5 (a) Condition 1 (b) Condition 2 (c) Condition 3 Figure 8. Plots of the length of disordered regions of super atoms vs. the number of super atoms on each condition 4. Considerations These fractal analyses show that self-similarity of the distribution of super atoms is considered to be held when N is larger than Nc (Nc = 250). Using this property and projection approach, it becomes possible to predict the actual arrangement of atoms. The region of analysis is assumed to be the average value of experimental values of trigger point for Ni-Cr-Mo-V steel, that is, 10 ~ 210 μm [10]. When actual distance between neighboring atoms is assumed to be 0.3 nm, the number of atom, N is considered to be 3×105. Based on Fig. 8, values of L and G at N = 3×105 were predicted as shown in Table 2. Table 2. Disordered region at N = 3.0×105 Direct method Verlet method Condition 1 Condition 2 Condition 3 Condition 1 Condition 2 Condition 3 L 0.238 0.384 0.0 0.211 0.359 0.0 G 0.203 0.062 0.398 0.183 0.055 0.420
13th International Conference on Fracture June 16–21, 2013, Beijing, China -9- These results show that the length of L and G are 20, 30 and 40 μm under three conditions, respectively. These values were almost equal to that of grain size for Ni-Cr-Mo-V steel and it is in good agreement with the dominating region of fracture obtained experimentally by Ritchie [8]. Therefore, our proposed method is found to well predict theoretically the dominating factor of fracture. 5. Conclusions (1) On the basis of the analytical method using the concepts of super atoms, super dislocation and self-similarity, the atomic distribution from super atoms to those with small scale was found to show the self-similarity and the possibility of analyzing the behaviors of actual atoms using projection method was validated. (2) The direct method and Verlet method were found to give the same results on the atomic distribution, however calculating time is different. (3) Using our proposed method, the dominating region of fracture was shown to derive and it was in good agreement with that obtained experimentally by Ritchie. References [1] G.R. Irwin, Handbuch der Physik, Springer 6 (1958) 551. [2] J.R. Rice, Int J Frac Mech, 2 (1966) 426. [3] T. Yokobori, An Interdisciplinary Approach to Fracture and Strength of Solids, 1968, Wolters-Noordhoff Pub. The Netherlands, T. Yokobori, Zairyo kyoudogaku, The 2nd Edition (1974) Iwanami shoten, In Japanese. [4] T. Yokobori, S. Sawaki, S. Nakanishi, Engng Fract Mech, 12 (1979) 125-141. [5] T. Yokobori, A.T. Yokobori, Jr., J of the Japan Institute of Metals, 42 (1978) 88-95. [6] A.T. Yokobori, Jr., T. Kuriyama, Y. Kaji, Advances in Fracture Research, Proc of ICF10, in the content of Special Lecture of CD-rom, Honolulu 2001, Elsevier Science (2001). [7] K.B. Broberg, Cracks and Fracture, Academic Press, (1999). [8] R.O. Ritchie, J.F. Knott, J.R. Rice, J Mech Phys Solids, 21 (1973) 395. [9] A.T. Yokobori, Jr., T. Ohmi, Y. Nagumo, D. Yoshino, Proc. of the 50th Japan National Symposium Strength, Fracture and Fatigue (2006) 57-60. [10] J. Watanabe, T. Iwadare, Y. Tanaka, T. Yokobori, Engineering Fracture Mechanics, 28, (1987) 589.
13th International Conference on Fracture June 16–21, 2013, Beijing, China -1- Numerical and analytical introduction to the prediction of grain boundary micro-crack initiation induced by slips bands impingement. Effect of material and microstructure parameters. Mohamed Ould Moussa1,*, Maxime Sauzay1 1 CEA, DEN, DANS, DMN, SRMA, F-91191Gif-sur-Yvette, France * Corresponding author: mohamed.ouldmoussa@cea.fr Abstract Micro-cracks are often observed at the intersections of thin slip bands (SB) and grain boundaries (GB) due to local stress concentration. Numerous models are based on the pile-up theory and the Griffith criterion, used since the pioneering work of Stroh. We have shown that the former underestimate strongly the macroscopic stress for GB micro crack nucleation. In fact, the key issue is that slip bands display finite thickness, observed to belong to [20nm 1000nm]. Therefore, one aims to account for the effect of SB thickness in crystalline finite element (FE) calculations performed using the Cast3M software. The simulations take into account the effects of isotropic elasticity parameters, cubic elasticity, GB orientation and crystallographic orientation of the considered grain. Following the theory of matching of asymptotic expansions, this leads to an analytical expression of the GB normal and shear stress, which show weaker stress singularities than the pile-up one. Keywords Micro-cracks, slip bands, pile-up theory, linear fracture mechanics, FE method, crystalline plasticity 1. Introduction Many issues are available dealing with the appearance and effects of either slip bands (SBs) or dislocation channels on the behavior of irradiated materials. Indeed, the intersection sites between SBs and grain boundaries (GBs) are prone to micro-crack nucleation because of strain localization. A series of papers [1 ; 2 ; 3 ; 4 ; 5 ; 6] highlight the presence of slip localization in Faced Centred Cubic (FCC) metals and alloys observed after post-irradiation tensile loading. Slip Bands were also observed in [7 ; 8 ; 9 ; 10 ; 11 ; 12 ; 13 ; 14] after cyclic loading. Such slip bands have been shown to be Persistent Slip Bands (PSBs) [9]. Some other works [15] have reported the formation of Slip Bands appearing during simple tensile loading. Whatever the loading conditions, such slip bands show a thickness lying between ten nanometers and a few micrometers, and a length about the grain size, usually varying from ten micrometers to a few hundred micrometers. Jiao et al. [5] have evidenced strong localization in austenitic stainless steels in post-irradiation tensile tests, using AFM measurements. Wejdemann and Pedersen [16] have applied the same techniques to observe such localization in the PSBs where plastic strain is shown to be fifty times larger than the macroscopic plastic strain. Sharp [1] and Edwards et al. [4] highlighted strain localization in single crystal and polycrystals of copper subjected to post-irradiation tensile loadings. Sauzay et al. [17] confirmed such localization in the case of irradiated austenitic stainless steels. In addition, several works attempted to model the stress concentration at grain boundaries. Besides, it is proved that the anisotropy character of crystalline elasticity induces stress concentration at grain boundaries according to Neumann [18]. Margolin and co-workers [19 ; 20] have carried out optical observations of slip traces and conclude that stresses are more concentrated near grain boundaries. The stress gradients around GBs, induced by plastic deformation incompatibilities between neighbor grains, can be tracked thanks to large-scale finite element (FE) computations [21].
13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- Such stress concentrations may allow to induce inter-granular crack initiation because they neglect plastic slip localization. Therefore, in the scope of the current contribution, one investigates the effect of localized slip on GB stress fields. An analytical approach, based on the well-known Stroh model [22], has been largely applied in order to evaluate GB stress. One recurrent issue has indeed been, since decades, the using of discrete or continuous dislocation pile-ups. The stress singularity, due to an edge or screw pile-up of length LPile-up has been shown to be the same as the one of a crack in the framework of linear elastic fracture mechanics (LEFM) [22 ; 23]. Therefore, an energy criterion has been proposed by [23], based on the Griffith criterion, for predicting micro-crack nucleation. The latter may be used for the case of a singularity exponent of 0.5 only and it fails when the exponent value is less than 0.5 because the energy release rate, G, becomes equal to zero [24 ; 25]. Cottrell [26] later suggested that the fracture process should be controlled by the critical crack growth stage under the applied tensile stress, which required higher stress than the crack nucleation itself as suggested by Stroh. Cottrell came with a modeling by supposing that slip occurs along two atomic planes which intersection shows pile-up dislocation. However, Sauzay and Evrard [17] recently have outcame with the limitation of such basic pile-up theory to more accurately predict micro-crack initiation. Following their work, the pile-up theory leads to an underestimation of macroscopic stresses with comparison to experimental data. Indeed, the pile-up approach postulates that slip localization occurs on one atomic plane only, but, experiments carried out using various materials and loading conditions show slip occurring on many slip atomic planes [8], [27], [5 ; 16], [28]. Therefore, because of the distribution of plastic slip through the slip band thickness [2 nm, 1000 nm], accounting for the SB thickness in the models may improve the predictions. The current contribution aims to validate an analytical model of GB stress fields in the case of SB impingement. Then, the first section presents the analytical modeling for close stress fields configuration which corresponds to points located near the intersection of the GB and the SB. The second section deals with finite element (FE) calculations in order to investigate numerically the effect of SB or GB geometry and material properties on GB stress fields. Besides, a final section is devoted, on one side, to adjust parameters and validate the analytical model and on the other side, to show the influence of GB and SB orientation on the adjusted model parameters and a conclusion will end the paper. 2. Analytical modeling The subsequent problem is to be solved: an elastoplastic slip band is embedded at the free surface of an elastic matrix, subjected to a displacement controlled tensile loading (Fig. 1a). a) b) Figure 1. a) Main grain, slip band and matrix b) SB-GB intersection and associated vectors SB GB n ρ m ρ GB n ρ θ x ρ y ρ
13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- Fig. 1a and b show the following parameters: - SB and GB orientation related ones: m ρ ( (1,0, 1) 2 1 − = ), n ρ ( (1,1,1) 3 1 = ), GB n ρ , SBα , GBα and θ denote respectively slip direction vector, normal vector to slip plane, normal vector to the GB, angle between slip plane and loading direction (m x ρ ρ , ), angle between the GB normal and loading direction (n x GB ρ ρ , ) and an angle between the SB and the GB, given by: θ GB SB o α α = − + 90 . (1) - SB size and loading parameters: t, L, 0Σ and f are respectively SB thickness, SB length, macroscopic applied tensile stress and Schmid factor. In addition, for further assumptions, let nnσ , nmσ , 0τ and r be respectively the GB normal stress, the GB shear stress, SB yield shear stress and the distance to the SB along the GB. It is worth to note that the developments involved in the current paper concern the close fields configuration which means that one focuses on stress evolution near the intersection of the SB and the GB (at a distance r such as 0 < r << t). The point located at the intersection of SB and GB corresponds to r = 0. GB stresses singularity is the same as the crack one in the LEFM framework, leading to an exponent of 0.5 of the stress expansion. [22 ; 23], the GB normal stress field induced by one edge dislocation pile-up is given by: ( ) ∞ − − Σ − +Σ = n pile up pile up n h f r L r ) ( ) / ( 2 3 ( , ) 0 0 1/2 θ τ θ σ , (2) where ( ) sin cos( / 2) θ θ θ = h , Σ = ∞ n cos ( ) 2 0 GBα Σ and /2 L L pile up = − . The absence of any term accounting for the SB thickness, t, in Eq. 2 is noticeable. Indeed, slip is assumed to occur on one atomic plane only as mentioned earlier. However, experimental observations have shown that slip may occur in many atomic plane and lead to the question of taking into account SB thickness. This implies the existence of two characteristic lengths: SB length and thickness, in the new problem of finite thickness. It is also proved [17] that the driving force ) ( 0 0τ Σ − f is proportional to the macroscopic shear stress. These two points make our problem be similar to the case of a crack with a V-noch tip in an elastic matrix even the stress singularity is induced by a slip localization in ours. That is why, following the theory of matching expansions [24 ; 29], we perform a modeling of the GB normal and shear stress close fields with respect to the SB length, L and the SB thickness, t: ( ) ( ) ) / ( / ( ) 0 0 0.5 τ σ α Σ − = f r A L t t r nn nn , (3) and ( ) ( ) ) ( / / ( ) 0 0 0.5 τ σ α Σ − = f r A L t t r nm nm . (4) α is the singularity exponent nn A and nm A are model parameters. The subscript “nn” corresponds to the GB normal stress and “nm” to the GB shear stress. It is worth to highlight that this model assumes a linear dependence of GB stresses on the driving shear stress, T 0 0 τ = Σ − f , and the same singularity exponent is valid for both shear and normal stress components and whatever L and t. The main difference between this model and the pile-up one is that the finite SB thickness, t, is taken into account. The stress singularity is assumed to be weaker in the proposed model than in the pile-up case, 0.5 <α as it will be probably shown.
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